400 
Proceedings of Royal Society of Edinburgh. [sess. 
On the Isoclinal Lines of a Differential Equation of 
the First Order. By J. H. Maclagan-Wedderburn. 
Communicated by Professor Chrystal. 
(Read January 5, 1903.) 
A differential equation may be regarded from two points of 
view, one purely analytical, the other geometrical. From the 
analytical point of view, a differential equation of the first order 
is merely a functional relation between x , y , and p (where 
p^dy/dx ), and the problem of solving the equation is to find a 
function of x, say f(x), such that if f(x) and df(x)/dx are substituted 
for y and p in the equation, the result is an identity in x. In the 
geometrical method, on the other hand, x and y are treated as the 
co-ordinates of a point in a plane and p as a direction. The 
differential equation then attaches to every point in the plane a 
certain direction, which may be conveniently represented by an 
arrow drawn through the point. The problem of integration then 
resolves itself into finding a family of curves, such that, at every 
point ( x', y'\ the direction of the curve at that point is the direction 
obtained by substituting x and y' in the differential equation and 
solving for p. These curves are called the integral curves of the 
equation. This method owes its development chiefly to Lie. 
An instructive example of a differential equation from this point 
of view is furnished by a well-known experiment in magnetism. 
A magnet exerts on another magnet, placed in its neighbourhood, 
a force whose direction and magnitude depend, in a given medium, 
solely on the strength of the two magnets and on their relative 
position, and, if one of the magnets is very small, the force on it 
due to the other is merely directive. We have here, then, a 
physical representation of a differential equation. If now we 
cover a magnet with a sheet of paper and sprinkle iron filings on 
it, each filing becomes a magnet by induction, and therefore sets 
itself longitudinally in the direction of the force at the point where 
it falls, and, if the paper is gently tapped, the filings arrange them- 
